The explicit local unitary evolution matrix for the grid pair is. We then apply the substitution rule to 1D free fermionic QCA system. Focusing on the grid pair , four states evolve as follows. Here and are fermion creation operators fulfilling anti-commutation. Namely, the local unitary evolution matrix is. It should be noted that if periodic boundary condition is adopted for the grid pair , the off-diagonal element in Equation 20 must be replaced with considering that etc. M is the total number of particles in all grid points.
Essentially same equation as Equation 20 is proposed in other literatures on multiparticle QCA   . Now we introduce an interaction between particles.
For this purpose, it is reasonable to introduce an additional phase rotation factor by the potential caused by other particles just like the external potential case. Note that QCA with external potential was firstly studied by Meyer . QCA with the nearest neighbor pair interaction was studied also by Meyer  and Boghosian  and Schumacher and Werner  in the form we present here.
Here we discuss the simplest case, namely the cases where the nearest neighbor interaction is included. We assume that interaction occurs as an additional phase rotation only when two particles exist in the neighboring grids. In the context of QLGA two-component QCA , this additional phase rotation corresponds to the phase shift by the collision between the left-going and the right going particles . Here meansattraction, and means repulsion. Note that in this simplest case, the structure of evolution scheme is kept same as the structure of free fermion case.
Here SVD is applied then the subspace corresponding to small singular values is truncated in order to keep the dimension of auxiliary space to the given value m in this case. Figure 4. This is an example of 8-grid system. A general wave function of the 2nd quantized QCA is represented by a rank-8 tensor.
Firstly this rank-8 tensor is approximated by the MPS form namely by the contraction of 8 low rank rank-3 or rank-2 tensors. Then the 2nd quantized QCA rule is applied upon the time evolution, namely the contraction with the tensors, four Us even time or three Us plus two odd time.
In this diagram the contraction is assumed to be performed for any connected pair of legs. Figure 5. Time evolution in the TEBD algorithm. Finally in this section, we compare our method with the ordinal way of reaching the TEBD algorithm. Basically so called hopping term representing kinetic energy part in evenly-spaced-grid-base or site-base quantum models such as Hubbard model or fermionized XXZ model is derived from the FDM-ap- proximation of kinetic energy term.
Here is the external potential at the position x. The 1st term is so called hopping term. By adding neighboring interaction term and dropping external potential term and constant term for simplicity, we have fermionized XXZ model   , where only nearest neighbor grid point of occupation number have a interaction through. If the anisotropy parameter , this model corresponds to the XXX model for fermion where the interaction is two-body coulomb interaction.
The XXZ spin model and its equivalent fermionized version are well studied . The phase diagram of the XXZ spin model in extended systems with the external magnetic field consists of 3 phases, ferromagnetic, paramagnetic and antiferromagnetic phases. When the magnetic field is zero, correspond to ferromagnetic gapped , paramagnetic gapless and antiferromagnetic phases respectively and in the paramagnetic phase, quasi particles magnon behave as boson-like Tomonga-Luttinger liquid  .
The magnetic field in the XXZ spin model becomes the chemical potential when the model is fermionized. The method has been used for grand canonical systems. We however focus our application in finite system where the number of particles fixed. Time evolution during the small time interval is done as follows Suzuki-Trotter. As terms in or commute with each other, we have. This is QCA-like evolution. As each factor is finite matrix, we can obtain easily its matrix representation using the standard matrix representation of creation and annihilation operator as follows.
We see the exact correspondence of the hopping term parameter in the model Hamiltonian to the QCA parameter , and the strength of correlation introduced by can be interpreted as the phase factor caused by the local potential at the grid point from the other electron in QCA. When , namely , it corresponds to the free Boson approximation case we will address in the next section. We propose here boson approximation by fermionic QCA or QCA with a hard core condition when occupation number per grid is small.
We mean by the hard core condition that at most one-particle can reside in one grid point. We not necessarily mean. We assume that only the amplitudes at points where all are different comprise the full set of independent variables and amplitudes of other points etc. We illustrate in Figure 6 the method of the boson approximation we propose for two-particle case, comparing with free fermionic QCA case. In order to make it easy to understand, we compare with the free fermion case, where no interpolation is needed, namely we set the amplitudes where to zero. For example, in 2-particle free fermion case, must be anti-.
Figure 6. We assume that for the fermion case, and for the boson approximation case. Note that even in the boson approximation case, is not an independent amplitude and it is interpolated from other points. And we can obtain the evolution rule at a quadrilateral on the diagonal line as follows.
For 2-particle Boson approximation case, must be symmetric with respect to exchange of , but the amplitude cannot be given without some assumptions. We take an approximation to assume that.
This implies, the 4 by 4 Unitary matrix in 2nd quantization formalism changed from that of Fermion case as follows. We give another possible interpretation of Equation The boson approximation Equation 34 can be obtained by applying coarse graining to Equation 18 using the following seemingly reasonable weight matrix for the adjacent grid pair subspace. Here, coarse graining means that three states of the adjacent grid pair, namely , are joined into one state 1,1 so that the occupation number per grid is kept less than 2 upon time evolution.
However, as shown in the Supplementary Material, Sec. Many-body effects on the transport properties of single-molecule devices. Genre The easiest way to find what you're looking for is to search by topic or interest. Arrives by Wednesday, Oct 2. Exponentially more precise quantum simulation of fermions in second quantization. Uber das Paulische Aquivalenzverbot. This development provides an alternative exact route to calculate the static and dynamical properties of fermionic systems and sets the stage to exploit the quantum-classical and semiclassical hierarchies to systematically derive methods offering a range of accuracies, thus enabling the study of problems where the fermionic degrees of freedom are coupled to complex anharmonic nuclear motion and spins which lie beyond the reach of most currently available methods.
Note that the original bosonic QCA Equation 18 does not conserve hard core condition due to the transition from to 0,2 or 2,0. In our simulations, we adopt the minimal auxiliary space dimension for MPS which can describe any 1-slater wavefunction namely for M particle system. For example, in 2-particle case, using the standard representation of Fermion creation and annihilation operators. For free fermion , it converges to the state where the two particles occupy the ground and the 1st-exited 1-particle states.
For free boson approximation , it converges to the state where the two-particle reside in the same 1-particle ground state. Similarly we computed MPS wave function for the three particle system and the results are shown in Figure 8. In general the parameter of the interaction must be scaled properly when the grid spacing is changed in order to obtain the same continuum limit waveform.
It is reasonable that when the ground state waveform does not depend on N, and when it depends on N. The case of is exceptional in that the waveform does not depend on N as if the particles were not interacting despite the fact that interaction is taken into account by non-zero parameter. This reflects the validity of the boson approximation.
Converged density distribution of two-particle system , zero amplitude boundary condition. Theoretically the ratio should be zero for the free fermion case , but small numerical error is observed. In Figure 10 , we show the converged density distribution of two-particle system. For the case, the discontinuity of the wave funcion can be seen at. In general the boson approximation wavefuncion and the real fermionic wavefunction are thought to be related by Equation 42 . Figure 8. Converged density distribution of three-particle system , zero amplitude boundary condition.
To perform imaginary time simulation, we set simply in the unitary matrix Equation 30 to the imaginary value. We performed a canonicalization of MPS state proposed by Vidal  at each simulation step, and in addition to this we performed an appropriate gauge transformation of MPS state corresponding to an additional evolution by the spatially constant chemical potential. In a MPS simulation of systems of fixed particle numbers, the chemical potential is theoretically irrelevant to the result, but it affects the robustness of the simulation and a small numerical error causes violation of particle number conservation leading to the grand canonical ground state.
Figure 9. The ratio of sum of the truncated norms of singular values to that of all singular values in the two-particle system. Of course in contrast to the above QCA-TEBD simulation, this can be performed only when the number of grids or particles is small notorious exponential wall for large number problems. But as it is simple and free from the particle number conservation problem, the result can be used as a reference to the QCA-TEBD simulation. Moreover there are no fundamental difficulties, in simulating bosonic or higher dimensional systems by the 1st quantized form.
We already explained the relation between the 1st quantized QCA and the 2nd quantized QCA in the free particles case, we here explain how to treat the additional phase rotation caused by interactions in the 1st quantized QCA. Firstly we explain the 1D-2 particle case. One step evolution is given by Equation Figure Top: , Middle: , Bottom:.
At an even or odd time the evolution rule Equation 43 is applied to each even or odd quadrilateral namely red or blue quadrilateral in Figure 6 respectively. Precisely the rule Equation 43 is for the bulk. At the zero boundaries the application of U in are partially replaced by the simple phase rotation of Equation 11 at an odd time. The algorithm of the 1st quantized form of QCA is basically independent of particle statistics. The only procedural difference between boson and fermion is in symmetrization or anti-symmetrization at each simulation step. Without this anti-symmetrization however a decay from a fermionic state to a bosonic state occurs occasionally.
In more general 1D M-particle case, in Equation 43 becomes. For example, at the point in 1D 4-particle case, the additional phase rotation at an even time is which comes from and. In higher dimensional case, the free evolution part is the same as in 1D case, and only the paring condition for the additional phase rotation need to be modified except for the obvious anti -symmetrization procedure. In a case of higher dimension or many particles, the requirement for the magnitude of becomes severe. If we set upper bound of phase rotation per.
Here D is the dimension of the. For the more practical programing, we reserve. In the following, we show two imaginary time 1st quantized QCA simulations, one is 1D 4 particle fermionic and corresponding bosonic system, the other is 2D 2 particle fermionic and bosonic systems. We show the result of 1D 4 particle and 2D 2particle imaginary time simulations in Figure 11 and Figure 12 respectively. More generally, by adding extra phase rotation caused by neighboring grid pair interaction, we conclude that.
Converged density distribution of the 1D 4 particle fermionic and bosonic imaginary time simulations. Upper left: the 2-particle reduced density of the 1D 4-particle bosonic system, Upper right: that of the corresponding bosonic systems, Lower: the 1-particle reduced density. Converged 1-particle reduced density distribution in the 2D 2-particle 1st quantized QCA simulations.
In the process of the time evolution, the bond axis rotates from the original. In the fermion case, around , there seems to be a transition point to the condensation. In Figure 11 we confirm that this approximation is very good for. Note that similar boson-fer-mion correspondence for 1D continuous space quantum system is well known, namely, the bosonic system with infinite repulsive delta function behaves as a free fermion  .
By seeing Figure 12 one might expect that fermionic 2D system with behaves approximately the same as the 2D free bosonic system , but in more than 1D system, there is no such a simple correspondence between fermion and boson as 1D system, because collision points are qualitatively different from boundary points in more than 1D system. For 1-body-reduced density distribution for bosonic system are well approximated by that of fermionic system when. But after exceeds 1 the error becomes rapidly larger. For 2-body- reduced density distribution, the error increases rapidly when reaches slightly below 1.
In the condensation state , the assumption used for the wave function interpolation seems to become inapplicable. In this supplementary section, we briefly discuss the possibility of multi-step QCA. Vertical axis indicates the magnitude of the difference between density distribution of the converged ground states for bosonic and fermionic systems.
This equivalence is easily shown by using the factorization form of the two-grid translationally invariant banded unitary matrix namely multi-step QCA form  . Namely their two components up and down are assigned to two amplitudes of adjacent grid points in the lattice of which the number of grid points are doubled from the original lattice.
Now we rewrite in a factorization form. The only difference between QW and DCA is the definition of the two-component up and down state which are indicated by ellipses. Considering we have. The expressions in the parentheses of Equations However in this case, obtained 1DDirac equation is not ideal one. In the QW case, the situation is the same. In the following we explain the outline of this situation. The case where has the form is particularly simple and important and we restrict our argument to this case.
In order to be able to connect this QCA with the Dirac equation, in the wave number expansion. Moreover it would be ideal if. Although actually in all cases by similar calculations, only in. Therefore the same arguments about the boundary condition, the 2nd quantiza- tion formalism, the simplest interaction and the boson-fermion corresponding as in QCA apparently hold.
In this study we show that in one-dimensional multiparticle QCA, the approximation of the bosonic system by fermion boson-fermion correspondence can be derived in rather a simple and intriguing way, where the principle to impose zero-derivative boundary conditions of one-particle QCA is also analogously used in particle-exchange boundary conditions. As a clear cut demonstration of this boson approximation, we calculate the ground state of 2 or 3-particle systems in a box using imaginary time QCA-TEBD simulation. Obtained ground states are indeed boson-like. We also perform imaginary time simulations by the 1st quantized form of QCA not only for fermionic system but also for bosonic system and show the applicable range of boson approximation boson-fermion correspondence.
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